MR Damper Modeling using Gaussian and Generalized Bell of

九州大学学術情報リポジトリKyushu University Institutional Repository

MR Damper Modeling using Gaussian andGeneralized Bell of ANFIS Algorithm

Choirunisa, IkhtiarMechanical Engineering Department, Universitas Sebelas Maret

UbaidillahMechanical Engineering Department, Universitas Sebelas Maret

Imaduddin, FitrianMechanical Engineering Department, Universitas Sebelas Maret

Elliza Tri MaharaniMechanical Engineering Department, Universitas Sebelas Maret

他

https://doi.org/10.5109/4491844

出版情報：Evergreen. 8 (3), pp.673-685, 2021-09. Transdisciplinary Research and EducationCenter for Green Technologies, Kyushu Universityバージョン：権利関係：

EVERGREEN Joint Journal of Novel Carbon Resource Sciences & Green Asia Strategy, Vol. 08, Issue 03, pp673-685, September 2021

MR Damper Modeling using Gaussian and Generalized Bell of ANFIS Algorithm

Ikhtiar Choirunisa1, Ubaidillah1,*, Fitrian Imaduddin1, Elliza Tri Maharani1, Gigih

Priyandoko2,*, Saiful Amri Mazlan3 1Mechanical Engineering Department, Universitas Sebelas Maret, Jalan Ir. Sutami 36A, Kentingan, Surakarta

57126, Indonesia 2Electrical Engineering, Universitas Widyagama Malang, Malang, Indonesia

3Malaysia Japan International Institute of Technology, Universiti Teknologi Malaysia, Malaysia

*Author to whom correspondence should be addressed: E-mail: [email protected], [email protected]

(Received January 7, 2021; Revised September 6, 2021; accepted September 6, 2021).

Abstract: The MR damper is well-known for its hysteresis characteristic, which needs to be

modeled accurately to describe the MR damper's original state. Modeling with conventional calculations is considered less effective for MR damper because it has a very high nonlinearity. One of the modeling methods chosen is Adaptive Neuro-Fuzzy Inference System (ANFIS), using two types of membership functions: Gaussian and Generalized Bell. Overall, the results showed that Gaussian had an accuracy about 1% higher than Generalized Bell. It can be considered in future studies to model the MR damper and designing a control system.

Keywords: MR Damper; Gaussian; Generalized Bell; ANFIS; Modeling

1. Introduction

The suspension system is useful for maintaining contact between the wheels and the road, and maintaining driving stability, so the vehicle remains under control 1). In a vehicle suspension system, one of them is called a semi-active device. A critical part of the semi-active device is a magnetorheological damper (MR damper) that changes the damping coefficient. MR dampers are used in industry or control structures such as in semi-active controls utilizing LQR controllers 2), industrial shock absorbers 3), and structural control 4). The MR damper has a difference from a standard damper that generally uses a fluid such as a hydraulic fluid because it is designed using a special fluid called magnetorheological fluids (MRFs). It has been classified as one of the smart materials since it is easy to control under external influence.

MRFs are magnetic suspensions that can change their rheological properties when subjected to the magnetic field induction 5). It consists of three main parts 6), namely soft magnetic particles (Fe3O4, Fe3 , Fe, or Co), carrier fluid or a soft magnetic particle suspension medium (synthetic oil or water and mineral oil), and the last is additives (a dispersant or an anti-sedimentation agent). MRFs are used in various mechanical system devices such as dampers, brakes, clutches, hydraulic valves, polishing devices 7), or semi-active vibration isolation 8). This fluid is referred to as a Non-Newtonian fluid when the magnetic

field is applied. After the magnetic field is removed, MRFs will be a Newtonian fluid 9). MRFs act as a nonlinear component in the MR damper, which causes the MR damper to have distinctive and well-known behavior, which is hysteresis characteristic 10). It needs to be modeled precisely and effectively to describe the typical behavior of MR damper 11).

Many modeling techniques have been used to model the MR damper, such as dynamic modeling 12). For example, a new phenomenological model based on the Bouc-Wen modeling proposed by Spencer et al. 13) in 1997. The results showed that the modeling was effective in controlling algorithm development and system evaluation. In 2005, Rene et al. 14) utilized a new mathematical model, namely the modified Lugre friction, to model the MR damper's dynamic behavior. The experimental data showed a fair degree of accuracy between the predicted and measured forces, although there were some minor differences at the low velocities. After two years in 2007, Kwok et al. 15) used the non-symmetric Bouc-Wen method identified by the Genetic Algorithm (GA) to consider the effects of the non-symmetric hysteresis behavior of MR damper, which was not considered in the original Bouc-Wen model. In 2011, Boada et al. 16) used the Recursive Lazy Learning method based on the Neural Network to model MR damper's behavior. This method reduced the previous compilation and used the sample input to guide making a decision. Arias et al. 17) modeled the MR damper

- 673 -

mailto:[email protected]


using a second-order polynomial model, and the results were used to design and apply a semi-active control algorithm. Dimock et al. 18) utilized Bingham's biplastic analysis to calculate shear thing and shear thickening MRF properties. Soltane et al. 19) proposed an adaptation of the accurately regulated Bingham model to reproduce the hysterical behavior of MR damper. Paul et al. 20) utilized a Neural Network model 21) consisting of 6 input neurons, 1 output neuron, and 12 neurons in a hidden layer to imitate the dynamic behavior of MR damper. Maharani et al. 22) and Han et al. 23) conducted experimental studies and mathematical modeling on MR damper.

However, these modeling methods are still inaccurately to model the hysteresis behavior of the MR damper. It is because of the high nonlinearity between the input and the output. Some modelings with conventional mathematical method are not suitable to describe the nonlinear state of the system. Because apart from being complex, the parameters are not known certainly. Complex systems require effective modeling and have a higher degree of accuracy. For this reason, the soft input technique is one of the main alternative options because it has been shown to be effective in dealing with complex problems and nonlinear behavior of control systems 24). Adaptive Neuro-Fuzzy Inference System (ANFIS) as machine learning 25,26) was chosen as alternative modeling. ANFIS is based on artificial intelligence, which combines fuzzy systems and neural networks 27,28) using hybrid learning 24). This method is proven to generate better results of modeling. ANFIS is used in modeling MR damper or in control structures by Nugroho et al. 10), Imaduddin et al. 29), Zeinali et al. 30), Zong et al. 31), Wang 32), and Nguyen et al. 33).

The efficiency and computaional cost of ANFIS is strongly influenced by the form of membership function and the number of membership function, so this study needs to be carried out. This study was expected to develop an accurate MR damper modeling using ANFIS with Gaussian and Generalized Bell membership function to describe the MR damper's hysteresis characteristics. So, the modeling results can be considered for designing a control system in the next future studies.

2. Research Methodology

The MR Damper prototype was manufactured at PT Kayaba Indonesia and was designed with an annular-radial-annular valve type. Furthermore, dynamic testing was carried out to measure this prototype's characterization using Shimadzu Selvopulser L-type Loading Frame with a 4830 controller. This machine works by regulating the electric current. It will calculate the damping force after getting the command to run from the exsiting program from the selvopulser (parameter values for the test are inputted in this program). After running the program, the damping force will be detected by the load cell sensor and then the value will be automatically recorded and displayed in the selvopulser

program on the computer. The experimental configuration can be seen in Figure 1. Its parameters were given at different amplitude variations of 4, 8, 10, 12, and 16 mm. Each amplitude was given a current of 0.25 A difference, starting from 0 to 0.75 A and a frequency of 1.5 Hz. 2.1 Fuzzy Inference System

FIS (Fuzzy Inference System) is also known as a fuzzy rule based on a system composed of five function blocks 34), namely: 1. A function block that contains several fuzzy if-then

rules is called a rule base. 2. A function block that defines the membership function

of a fuzzy set is called a database. 3. A function block that shows the inference operation in

the rule is called a decision making unit. 4. The function block that converts the cryptic input into

a degree of match with a linguistic value is called the fuzzification interface block.

5. The function block that converts the fuzzy output into cryptic input is called the defuzzification interface.

The basis of the FIS system is to use the expression of

the if-then logic to generate an output from a set of inputs that have been provided or desired. If-then logic is expressed in the "IF A THEN B" such as "if pressure is large, so volume is small" 34). The value of the fuzzy if-then rule depends on the selected number of membership function configurations from the system.

In the case of MR damper, the input data for modeling is current, displacement, and velocity. There are represented by variables I, d, and v. So, the if-then logic rule is written as Equation (1),

IF (I is A1) AND (d is B1) AND (v is C1)

THEN

f1 = p1I + q1d + r1v + s1

(1)

Fig. 1: Experimental set up

2.2 ANFIS Algorithm ANFIS (Adaptive Neuro-Fuzzy Inference System) is a

modeling technique that is composed of a multilayer network structure and utilizes fuzzy reasoning and neural

- 674 -


network learning 35), so the output of a fuzzy system with an exact value will be generated from a fuzzy input model of cryptic value 36). In ANFIS there is a hybrid learning algorithm 35) that is applied to update the parameters so that the final parameters used in the fuzzy system will be obtained. Two other methods, namely gradient descent method and least squares method 35) were utilized to adjust the premise parameter and the consequent parameter identifier, respectively. ANFIS is a multilayer network where each layer has calculations to solve the problems borne by each layer, as written here 36): 1. The first layer is for calculating the fuzzy

membership of the input data. 2. The second layer is to calculate the fitness of each

fuzzy rule. 3. The third layer is to calculate the normalized fitness. 4. The fourth layer is for calculating the output of each

fuzzy rule. 5. The fifth or last layer is to calculate the total output

from the fuzzy system. Talpur et al. in 2017 37) stated that choosing the number

of membership functions and the form of membership functions will affect ANFIS performance in terms of complexity and level of accuracy. The membership function is defined as a function that describes the level or degree of input data in a value range of 0 to 1 so that the difference in the quality of data can be clearly identified. The membership function is described in the form of a curve and is constructed by several parameters. In this study, the membership functions are used for MR damper modeling, which was Gaussian and Generalized Bell, as can be seen in Figures 2 and 3.

Fig. 2: Gaussian MF.

There are three inputs: current, displacement and velocity from the 20th cycle data measurement are used to generate the value of force as output. In the first layer, the formula of Gaussian and Generalized Bell is used to generate the premise parameters, that are shown in Equation (2) and (3),

f(I, d, v)= e-(x-ci) 2

2σi 2 (2)

f(I, d, v)= 1

1+ � x-ca � 2b

(3)

where, f (I, d, v) is the output of the first layer as a function of current, displacement, or current, while σi, ci, a, b, c are premise parameters.

Fig. 3: Generalized Bell MF.

The calculation results from the first layer will be calculated in the second layer using AND rules that are written in Equation (4) as follows,

where fi (I) is the output for the current input, fi (d) is the output for the displacement input, and fi (v) is the output for the velocity input. Each function's magnitude is influenced by the value of the premise parameters obtained from the first layer.

The third layer's output is the average of the multiplication between each input divided by the number of nodes. The formula, as shown in Equation (5),

O3,i= wi� = wi/∑ wini=1 (5)

where 𝑤𝑤𝑤𝑤�� is the relative rule weight of the i node.

The output from the fourth layer is calculated according to Equation (6),

O4,i=wi� gi=wi� (pi(I)+qi(d)+ri(v)+si) (6)

where 𝑝𝑝𝑝𝑝, 𝑞𝑞𝑝𝑝, 𝑟𝑟𝑝𝑝 and 𝑠𝑠𝑝𝑝 represent the values of the consequent parameters that are obtained after the training process is complete.

In this study, however, the output from the membership function was chosen to be constant, so that 𝑝𝑝𝑝𝑝 = 𝑞𝑞𝑝𝑝 = 𝑟𝑟𝑝𝑝 were assumed 0. Then, the consequent parameters in the fourth layer are represented only by the value of 𝑠𝑠1

O2,i=wi=fi(I)×fi(d)×fi(v) (4)

Gaussian-shaped,

P=[σ c]=[2 5]

Bell-shaped,

P=[a b c]=[2 4 6]

- 675 -


(𝑔𝑔𝑝𝑝 = 𝑠𝑠𝑝𝑝). So, the final output from the fourth layer can be written as Equation (7),

O4,i=wi� gi=wi� (si) (7)

The average output of the membership function in the

fourth layer is recalculated in the fifth layer as the final output, as shown in Equation (8),

O5,i =∑ wi�ni=1 gi= ∑ wi�n

i=1 (si)/∑ w1ni=1 (8)

Relative error value from ANFIS is calculated using

RMSE from Equation (9),

RMSE= √1/n ( ∑ (Zt� -Z) ^2 (9)

where n is number of testing sets, 𝑍𝑍𝑍𝑍�� is actual force, and 𝑍𝑍 is predicted force. 3. Results and Discussion 3.1 MR Damper Characterization

The data from measurement of the MR damper characteristics were taken from the 16th cycle of the entire MR damper testing cycle and generated a hysteresis relationship between force and velocity that can be seen in Figures 4 – 8. The damping force will increase when the electric current tested on the MR damper is also high as a result of the increased pressure drop that occurs in the valve 29). Besides the pressure drop, another factor affecting the damping force's magnitude is the high and low amplitude value. It corresponds to the theory of simple harmonic vibrations, which is written according to Equation (10),

Y=A sin (ωt) (10)

where Y is the relative displacement, A is the amplitude, ω is the natural frequency, and t is the time.

From Equation (10), the amplitude will affect the relative displacement of a vibration system. The derivative of displacement as a function of time (in this case is the relative velocity of the piston rod on the MR damper) will directly affect the damping force. Since the magnitude of the damping force is calculated according to the formula = 𝑐𝑐𝑐𝑐, c is the damping force coefficient, and v is the velocity.

Based on those figures, the damping force generated at amplitudes of 4 mm to 16 mm when on the on-state condition (0.75 A) was more significant than on the off-state condition (0 A). Also, the damping force for the overall amplitude of 16 mm had the most terrific value

compared to the others. This was according to the theory that the damping force is influenced by the damping coefficient and the piston rod's relative velocity, which is determined by the amplitude value. 3.2 The Results of MR Damper Modeling

MR damper modeling was done using ANFIS with two different membership functions with the number of epochs was 2500 and the same configurations of 3-5-6 that can be seen in Figure 9. The configurations in this study were selected based on the research that was done by Imaduddin et al. 29), while the analysis was done by Zeinali et al. 30).

In ANFIS, there is a term called Root Mean Square Error (RMSE), which describes the amount of data calculation error during the training process, and the results will appear when the training process is complete. The error value of the training process for Gaussian and Generalized Bell is shown in Figures 10 and 11, respectively, with the result of Gaussian was 0.1011 while Generalized Bell was 0.1079. From the graph, it can be seen that the error value generated between the two membership functions showed the results reaching 0.1. However, the level of accuracy of Gaussian was better than Generalized Bell because it showed an error value, which was 0.1011 and 0.1079, respectively. The number of epochs and fuzzy-rules was the same; it turned out that the training error value had a 0.0068 difference. The graph's trend showed that the calculated error value for Generalized Bell was saturated when it reached almost 1000 to 2500 epochs. When the number of iterations increased, then the error value remained the same. In addition to the error value, the differences in ANFIS training results with Gaussian and Generalized Bell are shown in Table 1.

Furthermore, the ANFIS training process generated parameters representing all training data and are known as premise parameters and consequent parameters. The premise parameter determines the degree of membership from the training data, while the consequent parameter determines the membership function's output value. These parameters are formulated in determining the value of an output from the modeling results. Tables 2 - 4 show the premise parameters' values and consequent parameters for Gaussian and Generalized Bell membership function. Based on the tables, the number of premise parameters represents the membership function configuration's input value, namely 3-5-6. Meanwhile, the number of consequent parameters is the product of the membership function's configuration as a fuzzy if-then rule.

In the scope of MR damper modeling, one of the useful parameters that can be chosen to predict the damping force using ANFIS is velocity, as described by Zeinali et al. 30). Figures 12 – 14 show the force's prediction versus velocity in off-state condition at 0 A and on-state at 0.75 A with the amplitude variation for 4, 10, and 16 mm. Based on the figures, the Gaussian and Generalized Bell membership

- 676 -


functions successfully predicted the peak value of force-velocity.

Fig. 4: Plot of force-velocity for 4 mm





Fig. 9: ANFIS structure for 3-5-6 MF.

I

d

v

f

- 677 -


Fig. 10: Gaussian RMSE

Fig. 11: Generalized Bell RMSE

Fig. 12: MR Damper modeling results on the 16th cycle test in off-state (0 A) and on-state for 4 mm

Fig. 13: MR Damper modeling results on the 16th cycle test in off-state (0 A) and on-state for 10 mm

Fig. 14: MR Damper modeling results on the 16th cycle test in

off-state (0 A) and on-state for 16 mm

Table 1 ANFIS information for Gaussian and Generalized Bell

ANFIS

Informations

Gaussian Generalized Bell

Epochs 2500 2500

Number of nodes 214 214

Number of linear

parameters

90 90

- 678 -


Number of nonlinear

parameters

28 42

Number of fuzzy-

rules

90 90

Value of training

error

0,1011 0,1079

Table 2 Premise parameters for Gaussian MF.

Input MF. σ c

Current 11 0.039756 -0.004153

12 0.604219 -0.351618

13 0.320881 0.72403

Displacement 21 5.098082 -

20.57685

1

22 4.499525 -8.989566

23 1.36592 -0.782151

24 2.643117 5.528595

25 2.119668 8.575908

Velocity 31 30.193096 -

151.0422

84

32 38.732401 -87.4658

33 22.121275 -

35.60300

5

34 22.4923 9.701959

35 37.153932 64.40899

36 45.287408 140.3465

63

Table 3 Premise parameters for Generalized Bell MF.

Input M

F

A b c

Current 11 0.002286 2.00917

4

-8.66E-05

12 0.33519 1.95733

7

0.197528

13 0.211905 2.01876

6

0.602006

Dsiplacemen

t

21 4.26646 1.79238

2

-16.611529

22 4.094669 3.59171

8

-7.706637

23 2.944155 -

0.16580

8

0.100385

24 4.848748 6.88285

9

6.831874

25 6.871987 3.79629 13.540689

Velocity 31 32.69824

4

1.27020

5

148.67971

7

32 33.15936

4

1.17714

6

-90.40958

33 29.43965

9

0.75816 -35.797875

34 30.15750

7

-

0.04668

3

30.09459

35 40.25923

2

1.61227

6

81.067577

36 30.63752

8

1.14221

1

150.46895

6

Following the theory, the damping force generated from ANFIS modeling using Gaussian and Generalized Bell was directly proportional to the increase in amplitude and electric current.

However, there was an exception for 10 mm because one of the velocity-damping force's peak value was not predicted correctly. It was caused by the small difference value of the displacement from 8 to 10 mm and 10 to 12 mm, while the other displacement parameters had a 4 mm difference value, namely 4, 8, 12, and 16 mm. Still, Gaussian and Generalized Bell membership functions were able to map the damping force, which was almost the same as the test result data. The evaluation results of the relative error values to the 16th cycle data testing are shown in Table 5.

Furthermore, it can be determined that each membership function's level of accuracy is compared with the average relative error values in the same current

- 679 -


conditions. In the off-state condition (0 A), Gaussian and Generalized Bell generated an average relative error value of 7.45% and 7.71%, respectively. The same phenomenon also occurred in the on-state condition (0.75 A) for Gaussian and Generalized Bell, which generated an average relative error value of 12.14% and 13.20%. From there, it can be observed that the average relative error values generated by Gaussian both in off-state and on-state conditions had a higher level of accuracy than Generalized Bell. A lower relative error value evidenced this fact. The same result applied to currents of 0.25 and 0.5 A.

- 680 -


Table 4 Consequent parameters for Gaussian and Generalized Bell

If-then

Rules

Consequent Parameters If-then Rules Consequent Parameters

Gaussian Generalized Bell Gaussian Generalized Bell

1 -8.793647698 -25.81086649 46 2.19384411 6.12074608

2 -2.29320457 -0.4886648 47 0.817232405 -0.345759205

3 -1.748439591 -3.046028459 48 1.620747585 0.052241079

4 -1.673680452 2.935599502 49 -1.353341094 -4.125143259

5 -1.528027213 -3.245283371 50 -1.308399275 -4.311688414

6 -2.051516492 -3.80859061 51 0.103466957 0.142749218

7 -2.125132976 -4.136189761 52 1.335350971 2.490750721

8 -0.716557057 -2.459437291 53 0.857580462 0.767886206

9 -0.512852651 -2.065684941 54 1.644383943 1.559701725

10 -0.075548291 4.70334813 55 -1.732805111 -3.981187687

11 -0.584024313 -2.839468931 56 -0.192948313 0.689588443

12 -0.161818266 -2.498484743 57 0.682806556 2.483973398

13 -1.869149739 -0.203973824 58 0.894222137 0.028510324

14 -0.718483185 1.236860981 59 1.033172705 1.626058501

15 -0.099990276 1.764036176 60 1.762638645 3.628203913

16 -1.664230426 -4.256253594 61 -5.362076276 -15.0646137

17 -0.513686034 1.156332972 62 -4.249359434 2.38346904

18 -0.048867155 1.846023437 63 -2.794574442 1.819955267

19 -2.139455656 -3.106856581 64 -4.58750393 -14.82540668

20 -0.031140378 0.721223673 65 -0.717376086 5.827991202

21 -1.823211229 -3.816942996 66 2.051742022 14.9858533

- 681 -


22 -0.896382293 1.705572623 67 -3.82051117 -1.468492107

23 -0.37054444 -0.977036973 68 -2.926865778 -0.627888446

24 -0.060391567 -0.793907304 69 -2.626727088 -0.207408557

25 -0.149420276 1.787449994 70 -3.149799095 -8.790917453

26 -1.320077109 -4.155640305 71 1.304391259 6.246806869

27 -0.919095751 -2.567454311 72 1.82939483 6.616388278

28 -0.242388313 2.245042636 73 -3.646147508 -7.061133014

29 -0.573313368 -1.739068096 74 -2.821149597 -6.231396294

30 -0.102770496 -0.545103018 75 -3.194661334 -5.88208006

31 -3.90276916 -11.83591939 76 1.140568842 5.867730054

32 -1.733557774 2.237976863 77 1.231250807 0.398225532

33 -1.090365173 1.721869009 78 1.954736388 0.407716387

34 -2.278949931 -10.24892461 79 -3.320124696 -6.447846344

35 0.923475591 5.178862868 80 -3.31725402 -6.308970276

36 -0.545681664 6.880643753 81 0.290034143 0.673140517

37 -1.815589925 -1.000446862 82 1.03219264 2.718555186

38 -0.990986112 -0.351345334 83 1.301484217 1.050095046

39 -0.852168765 0.142502653 84 1.950669566 1.772475946

40 -0.722267202 -5.323496547 85 -4.73095417 -7.801847737

41 1.231421939 4.526313185 86 0.022345596 1.499529935

42 1.386741155 4.74104485 87 0.346484995 2.304890225

43 -1.881660815 -5.6554021 88 1.491668348 1.888206377

44 -0.798857651 -4.342847263 89 1.310564375 1.056642497

45 -1.390561544 -4.500251447 90 1.984501282 2.623060161

- 682 -


Table 5 Model relative error to the 16th cycle measurement data Amplitudes Gaussian Relative Error (%) Generalized Bell Relative Error (%)

0.0A 0.25A 0.50A 0.75A 0.0A 0.25A 0.50A 0.75A

4 mm 8.93 11.32 10.61 11.84 9.12 12.48 10.88 12.27

8 mm 6.43 10.30 11.12 11.52 7.60 9.86 13.16 13.05

10 mm 9.96 9.83 10.93 14.54 10.24 8.61 10.92 15.53

12 mm 7.32 8.95 12.13 13.13 6.72 10.03 13.70 14.14

16 mm 4.62 7.37 8.48 9.69 4.88 7.22 9.27 11.04

Averages 7.45 9.55 10.65 12.14 7.71 9.64 11.58 13.20

Besides the current, another parameter to determine the quality of Gaussian against Generalized Bell compared at the relative error value when the same amplitude was applied. For 4 mm, the relative error value in Gaussian for each electric current started from 0; 0.25; 0.5; and 0,75 A was 8.93; 11.32; 10.61; and 11.84%. Consequently, Generalized Bell generated relative error values of 9.12, 12.48, 10.88; and 12.27%. When the amplitude was 16 mm, the relative error for Gaussian was 4.62; 7.37; 8.48; and 9.69%, while Generalized Bell is 4.88; 7.22; 9.27; and 11.04% (for each current from 0 – 0.75 A). These results prove that Gaussian had a higher level of accuracy than its counterpart (Generalized Bell). Similar results also occurred at amplitudes of 8, 10, and 12 mm.

Besides that, Gaussian has the advantage of distinguishing data quality, for example, small or large, high or low 37). A study by 38) proven that Gaussian has the best RMSE compared to others. It is because Gaussian can plot curves smoothly 37,38) and non-zero at any point 37). Based on the table, the relative error values will increase when the amount of electric current also increases. For example, at an amplitude of 16 mm, the relative error value for Gaussian in off-state condition (0 A) was 4.62% while in on-state condition (0.75 A) was 9.69%. The phenomenon also true for Generalized Bell, which generated the relative error value at 4.88% in the off-state condition and 11.04% in the on-state condition. These results were different from what has been studied by 29). The study stated that the greater the value of electric current and frequency, the smaller relative error values generated, and the relative error values began to consistent above the 1 Hz frequency. However, it was further explained that there are no clear facts about this phenomenon. Most likely due to anomalies originating from the data test, which require the same study with the

other data from different input and MR damper type 29).

Overall, the relative error value generated by the two ANFIS models was under 20%, and Gaussian showed an accuracy level is around 1% higher than Generalized Bell, with the maximum relative error was about 14.54% and 15.53%. These results corresponded to the value of the training error, which shows that Gaussian was smaller than Generalized Bell, so the accuracy of Gaussian was higher than Generalized Bell. It can occur due to a difference in the RMSE value generated during the training process. If the RMSE value is small, then the more accurate the calculations during modeling will be. In 2015, Gaxiola et al. 38) conducted a study to compare several types of membership functions namely Triangular, Gaussian, Trapezoidal, and Generalized Bell. These results from the study will be used to predict cases of Mackey-Glass time series. It showed that the best average prediction error (APE) was obtained by Gaussian, which was 0.0659. In addition, Ali et al. from 39) proved that Gaussian was suitable and performs well when used in systems dealing with both probability and statistics. Although Gaussian was only formed by two parameters (σ and c) 37), but this membership function has advantage can precisely, accurately, and clearly of distinguishing data quality, for example small or large, high or low 40). Still, study by 37) proven that Gaussian has the best RMSE compared to others. It is because Gaussian is able to plot curves smoothly 37,40) and concise notation so it is able to represent data effectively and non-zero at any point 40).

4. Summary

The MR damper was modelled using ANFIS with two membership functions, namely Gaussian and Generalized Bell. Each membership function was given the same ANFIS training parameters to generate the best results

- 683 -


from one proposed model. This study indicated that the training error value of Gaussian was 0.1011, while Generalized Bell was 0.1079. These membership functions generate training error value was close to 0.1. However, Gaussian showed an accuracy level of about 1% higher than Generalized Bell. Also, Gaussian has the best achieve than its counterparts because Gaussian can plot curves smoothly and concise notation to represent data effectively and non-zero at any point, although two parameters only form Gaussian. The evidence of its accuracy was from the maximum relative error value tested from one of the MR Damper test cycles in off-state and on-state conditions, which was 14.54% for Gaussian and 15.53% for Generalized Bell. Thus, it can be used as an MR damper modeling consideration in future studies' control system.

Acknowledgements Authors would like to acknowledge Universitas Sebelas Maret for financial support through Hibah Grup Riset Penelitian 2021.

Nomenclature

n number of testing sets (–) 𝑍𝑍𝑍𝑍�� actual force (kN) Z predicted force (kN) Y relative displacement (mm) A amplitude (mm) ω natural frequency (Hz) t time (s)

References 1) S. Ebrahimi-nejad, M. Kheybari, S. Vahid, and N.

Borujerd, “Multi-objective optimization of a sports car suspension system using simpli fi ed quarter-car models,” Mech. Ind. 21, 412 (2020). doi:doi.org/10.1051/meca/2020039.

2) G. Kumar, A. Kumar, and R.S. Jakka, “M echanics an adaptive lqr controller based on pso and maximum predominant frequency approach for semi-active control scheme using mr damper,” Mech. Ind. 19, 109 (2018). doi:doi.org/10.1051/meca/2018018.

3) A.M. Ã, and M. Hauke, “Application of magnetorheological fluid in industrial shock absorbers,” Mech. Syst. Signal Process., 28 528–541 (2012). doi:10.1016/j.ymssp.2011.11.008.

4) Z.Q. Gu, and S.O. Oyadiji, “Application of mr damper in structural control using anfis method,” Comput. Stucrtures, 86 427–436 (2008). doi:10.1016/j.compstruc.2007.02.024.

5) Z. Jian, Z. Jin-qiu, and J.I.A. Jin-feng, “Characteristic analysis of magnetorheological fluid based on

different carriers,” J. Cent. South Univ. Technol., 15 252–255 (2008). doi:10.1007/s11771.

6) X. Biao, L. Yiping, and R. Hongjuan, “Review on magneto-rheological fluid and its application,” Am. J. Nanosci. Nanotechnol. 2014, 2 (4) 70–74 (2014). doi:10.11648/j.nano.20140204.12.

7) Y. Bazargan-lari, “M echanics design and shape optimization of mr brakes using nelder – mead optimization algorithm,” Mech. Ind. 20, 602 (2019). doi:doi.org/10.1051/meca/2019017.

8) J. Xu, X. Yang, W. Li, J. Zheng, Y. Wang, and M. Fan, “M echanics research on semi-active vibration isolation system based on electromagnetic spring,” Mech. Ind. 21, 101 (2020). doi:doi.org/10.1051/meca/2019048.

9) C.J. Poynor, “Modelling and testing of mr dampers,” (November) 1–12 (2001). doi:10.13140/RG.2.1.2127.5604.

10) P.W. Nugroho, W. Li, H. Du, G. Alici, and J. Yang, “An adaptive neuro fuzzy hybrid control strategy for a semiactive suspension with magneto rheological damper,” Adv. Mech. Eng., 11 (2014).

11) M. Rahman, Z.C. Ong, S. Julai, M. Ferdaus, and R. Ahamed, “A review of advances in magnetorheological dampers : their design optimization and applications *,” J Zhejiang Univ-Sci A (Appl Phys Eng), 18 (12) 991–1010 (2017).

12) J. Ko, N. Takata, K. Thu, and T. Miyazaki, “Dynamic modeling and validation of a carbon dioxide heat pump system dynamic modeling and validation of a carbon dioxide heat pump system,” Evergreen, 7 (2) 172–194 (2020). doi:doi.org/10.5109/4055215.

13) B.F. Spencer, S.J. Dyke, M.K. Sain, and J.D. Carlson, “Phenomenological model for magnetorheological dampers,” J. Eng. Mech., 123 (3) 230–238 (1997). doi:10.1061/(ASCE)0733-9399(1997)123:3(230).

14) R. Jiménez, and L. Álvarez-Icaza, “LuGre friction model for a magnetorheological damper,” Struct. Control Heal. Monit., 12 (1) 91–116 (2005). doi:10.1002/stc.58.

15) N.M. Kwok, Q.P. Ha, M.T. Nguyen, J. Li, and B. Samali, “Bouc-wen model parameter identification for a mr fluid damper using computationally efficient ga,” ISA Trans., 46 (2) 167–179 (2007). doi:10.1016/j.isatra.2006.08.005.

16) M.J.L. Boada, J.A. Calvo, B.L. Boada, and V. Díaz, “Modeling of a magnetorheological damper by recursive lazy learning,” Int. J. Non. Linear. Mech., 46 (3) 479–485 (2011). doi:10.1016/j.ijnonlinmec.2008.11.019.

17) M. Arias-Montiel, K.H. Florean-Aquino, E. Francisco-Agustin, D.M. Pinon-Lopez, R.J. Santos-Ortiz, and B.A. Santiago-Marcial, “Experimental characterization of a magnetorheological damper by a polynomial model,” Proc. - 2015 Int. Conf. Mechatronics, Electron. Automot. Eng. ICMEAE 2015, (November) 128–133 (2015).

- 684 -


doi:10.1109/ICMEAE.2015.31. 18) G.A. Dimock, J.H. Yoo, and N.M. Wereley, “Quasi-

steady bingham biplastic analysis of electrorheological and magnetorheological dampers,” J. Intell. Mater. Syst. Struct., 13 (9) 549–559 (2002). doi:10.1106/104538902030906.

19) J. a C. Ambrósio, M.A. Neto, and R.P. Leal, “Mechanics of materials and structures october 2007,” J. Mech. Mater. Struct., 2 (October) 1515–1524 (2007). doi:dx.doi.org/10.2140/jomms.2015.10.195.

20) C.C. Chang, and P. Roschke, “Neural network modeling of a magnetorheological damper,” J. Intell. Mater. Syst. Struct., 9 (9) 755–764 (1998). doi:10.1177/1045389X9800900908.

21) H. Kumar, “Optimising parameters of fused filament fabrication process to achieve optimum tensile strength using artificial neural network optimising parameters of fused filament fabrication process to achieve optimum tensile strength using artificial neural network,” Evergreen, 7 (3) 373–381 (2020).

22) E.T. Maharani, U. Ubaidillah, F. Imaduddin, K.M. Wibowo, D. Utami, and S.A. Mazlan, “A mathematical modelling and experimental study of annular-radial type magnetorheological damper,” Int. J. Appl. Electromagn. Mech., Preprint 1–18 (2021). doi:10.3233/JAE-201560.

23) Y. Han, L. Dong, and C. Hao, “Experimental analysis and mathematical modelling for novel magnetorheological damper design,” Int. J. Appl. Electromagn. Mech., 59 367–376 (2019). doi:10.3233/JAE-171146.

24) M. Braz-César, and R. Barros, “Optimization of a fuzzy logic controller for mr dampers using an adaptive neuro-fuzzy procedure,” Int. J. Struct. Stab. Dyn., 17 (5) (2017). doi:10.1142/S0219455417400077.

25) T.G. Pratama, R. Hartanto, and N.A. Setiawan, “Machine learning algorithm for improving performance on 3 aq-screening classification,” Commun. Sci. Technol., 4 (2) 44–49 (2019).

26) G.D. Nugraha, G.D. Nugraha, B. Sudiarto, and K. Ramli, “Machine learning-based energy management system for prosumer machine learning-based energy management system for prosumer,” Evergreen, 7 (2) 309–313 (2020). doi:doi.org/10.5109/4055238.

27) N. Tarek, “System identification for quad-rotor parameters using neural network system identification for quad-rotor parameters using neural network,” Evergreen, 3 (1) 6–11 (2016). doi:doi.org/10.5109/1657380.

28) H. Han, M. Hatta, and H. Rahman, “Smart ventilation for energy conservation in buildings smart ventilation for energy conservation in buildings,” Evergreen, 6 (1) 44–51 (2019).

29) F. Imaduddin, S.A. Mazlan, Ubaidillah, M.H. Idris, and I. Bahiuddin, “Characterization and modeling of a new magnetorheological damper with meandering

type valve using neuro-fuzzy,” J. King Saud Univ. - Sci., 29 (4) 468–477 (2017). doi:10.1016/j.jksus.2017.08.012.

30) M. Zeinali, S.A. Mazlan, A.Y. Abd Fatah, and H. Zamzuri, “A phenomenological dynamic model of a magnetorheological damper using a neuro-fuzzy system,” Smart Mater. Struct., 22 (12) (2013). doi:10.1088/0964-1726/22/12/125013.

31) L.H. Zong, X.L. Gong, C.Y. Guo, and S.H. Xuan, “Inverse neuro-fuzzy mr damper model and its application in vibration control of vehicle suspension system,” Veh. Syst. Dyn., 50 (7) 1025–1041 (2012). doi:10.1080/00423114.2011.645489.

32) H. Wang, “Modeling of magnetorheological damper using neuro-fuzzy system,” 2 1157–1164 (2009).

33) S. Materials, S.D. Nguyen, and S.B. Choi, “A new neuro-fuzzy training algorithm for identifying dynamic characteristics of smart dampers,” Smart Mater. Struct., 21 (November 2014) (2012). doi:10.1088/0964-1726/21/8/085021.

34) J.-S. Roger Jang, “ANFIS : adaptive-network-based fuzzy inference system,” IEEE Trans. Syst. Man. Cybern., 23 (3) (1993).

35) F. Chang, and Y. Chang, “Adaptive neuro-fuzzy inference system for prediction of water level in reservoir,” Adv. Water Resour., 29 1–10 (2006). doi:10.1016/j.advwatres.2005.04.015.

36) H. Wang, and H. Hu, “The neuro-fuzzy identification of mr damper,” 6th Int. Conf. Fuzzy Syst. Knowl. Discov. FSKD 2009, 6 (1) 464–468 (2009). doi:10.1109/FSKD.2009.545.

37) N. Talpur, M.N.M. Salleh, and K. Hussain, “An investigation of membership functions on performance of anfis for solving classification problems an investigation of membership functions on performance of anfis for solving classification problems,” IOP Conf. Ser. Mater. Sci. Eng., 226 (2017). doi:10.1088/1757-899X/226/1/012103.

38) F. Gaxiola, P. Melin, and F. Valdez, “Comparison of neural networks with different membership functions in the type-2 fuzzy weights,” Adv. Intell. Syst. Comput., 322 707–713 (2015). doi:10.1007/978-3-319-11313-5.

39) O.A.M. Ali, A.Y. Ali, and B.S. Sumait, “Comparison between the effects of different types of membership functions on fuzzy logic controller performance,” Int. J. Emerg. Eng. Res. Technol., 3 (3) 76–83 (2015).

40) S.M. Azimi, “Designing programmable current-mode gaussian and bell-shaped membership function,” Analog Integr. Circuits Signal Process., (2019). doi:10.1007/s10470-019-01567-y.

- 685 -

Documents

MR Damper Modeling using Gaussian and Generalized Bell of